Uniqueness of K-polystable degenerations of Fano varieties

Abstract

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.

K-stability of birationally superrigid Fano varieties

“…In other words, the moduli space (if it exists) of birationally superrigid Fano varieties is separated. A similar statement is also conjectured for families of K-stable Fano varieties (postscript note: this has now been proved by Blum and Xu) and our proof of Theorem 1.7 is indeed inspired by the recent work [BX18] in the uniformly K-stable case. One should also note that if the answer to Question 1.5 is positive, then Theorem 1.7 follows immediately from [Che09, Theorem 1.5].…”

Positivity of the CM line bundle for families of K-stable klt Fano varieties

“…The reason for assuming in Theorem 1.20 that X t , t is uniformly K -stable for general geometric fibers, but setting δ to be δ X t , t only for very general geometric fibers is technical. On one hand, uniform Kstability is known to be an open property by [21,Thm A], and hence one may assume it on the general geometric fiber without imposing any additional assumption. On the other hand, only the function t → min{1, δ X t , t }, but not t → δ X t , t itself, is known to be constructible [22,Prop 4.3].…”

Reductivity of the automorphism group of K-polystable Fano varieties

“…Flat proper families X → S, whose geometric fibers are K-semistable Q-Fano varieties of dimension n and volume V , satisfying Kollár's condition (see [BX19,§1])…”

“…The ingredients needed in the construction can be translated into deep properties of such Fano varieties. See [BX19,Introduction] for a more detailed discussion of the prior state of the art.…”